UNIQUENESS THEOREMS FOR CONVOLUTIONTYPE EQUATIONS
D. J. NEWMAN
Of frequent appearance in the literature are those integral equations which take the form (1), below. These are commonly called
convolution-type
equations.
(1)
XF(x)- f F(t)K(x -t)dl = Gix)
for x G S.
J s
In particular, when 5 is the whole real line we obtain the standard
convolution equation. Again when 5 is the half line (0, <*>),we obtain
the Wiener-Hopf equation. S= [0, l] is still another of the classical
equations, known to aerodynamicists
as the "lifting line equation."
We will be concerned with the uniqueness question for the equation
(1), but in the following special sense: We wish to determine conditions on X and the kernel function K, together with class conditions
on K and P, which will insure the uniqueness of the solution of (1)
for all (measurable) sets 5.
For each fixed S, uniqueness is equivalent to the statement:
(2)
If XF(x)= f F(t)K(x - t) dt for all x G 5 then F(x) = 0.
•'s
Thus, if we redefine F to be 0 in the complement
(3)
XF = F*K iorxES,
And the logical conjunction
of S, (2) becomes
F = 0 for x <$5 =» F = 0.
of the statements
(3) for all (measura-
ble) sets 5 is simply:
(4) If, for each x, either F(x) = 0 or XF(x) = (F*X)(x), then F(x) s 0.
It is thus our task to find conditions under which (4) holds.
We choose as our setting an arbitrary locally compact abelian
group G with Haar measure dt. The class conditions will be Kit)
EL^G), F(0GPM(G).
Now let P(£) be the Fourier transform of Kit). As £ varies through
the elements of G, P(£) traces out a point set in the complex plane.
Received by the editors October 8, 1962 and, in revised form, October 26, 1963.
629
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630
D. J. NEWMAN
[August
We call this point set Ck, and the closed convex hull of Ck we call
HK.
Theorem.
Let K(t) have compact support. //XG#x
then (4) holds.
Of course, by the previous discussion, this theorem can be restated
directly in terms of the original uniqueness question. This restatement can be given as follows:
Theorem. Let K(t) have compact support and let \EHkFor any
(measurable) set S and any function G(x), the integral equation (l) has
at most one solution.
We apologize for the onerous restriction to K of compact support.
It is undoubtedly an unnecessary one. For G the real line, it can be
relaxed to the condition that K falls off exponentially.
It may even
be that no growth condition whatever on K is needed! The author is
simply unable to decide this question.
The condition expressed in this theorem is by no means best possible, but there is a kind of converse in the case of a Hermitian kernel
(i.e., one where K( —t) =K(t)*).1 This converse reads: If (4) holds,
X?¿0, and G is connected, then X$Hj.
For proof, note that K(i;) is continuous and real valued so that Ck
is a connected real set, and that Hk=Ck
or Hk=Ck^0.
In either
case, since Xf^O, }iEHK=^XECK=^K = foX(t)K(t)
acter X(t).
If this were so, the choice F(t)=X(
dt for some char—t) would contra-
dict (4).
In other cases the actual necessary and sufficient conditions to
insure (4) seem very difficult to obtain. Almost the simplest example,
that of the group of the integers, already gives (interesting) trouble.
If K consists of a single mass point, then our condition is necessary
and sufficient. Suppose, however, that we define our kernel, K(n), as
follows:
K(\)= -2, K(2)= -1, K(n)=0
for all other ra, and restrict our
attention to positive X.
Accordingly, the condition, XEHk, becomes the condition X>3/2.
It can be shown, however, that the actual necessary and sufficient
condition for (4) is the condition X> y/2 !
From now on we normalize by setting X = 1 and before turning to
our proof we would like to present the heuristic argument on which
it is based. This heuristic argument is actually rigorous in the case
of a compact G but is whimsical otherwise since we make use of the
"Fourier transform"
of F(t).
We use z* to denote the complex conjugate
of z.
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«
1965]
THEOREMS FOR CONVOLUTION-TYPEEQUATIONS
We are given that F(x)[F(x) —(F*K)(x)] = 0 identically.
JaF(x) [F(x) —(F*K) (x) ] dx = 0. Applying Parseval's
631
Thus
theorem we ob-
tain, then, JgHO*[H6 -P(Ç)£(£)] d£ = ° (where F(£) is the Fourier
transform of F(t)). Hence,
we conclude that
unless F(%) vanishes
a.e. (i.e., F(t)^0),
f |*(ö \»Ê(Q
di
J G
-
= 1,
(A Hol2dt
J G
and this shows that 1 EHrWe now turn to the somewhat intricate "truncation"
process
which supplies the rigor to these heuristic arguments.
Proof of Theorem. We assume that the hypotheses hold, viz.,
that 1<$.HK,that |F| £1, and that | F\ 2= F(F*K) identically. We
can express the first of these conditions
number a such that
(5)
Re[a(l - £({))] ^ 1
by the existence of a complex
for all { G G.
Now let XoEG be an arbitrary point and let F be a symmetric compact set containing 0, a neighborhood of x0, and the support of K.
We denote by V", as usual, the set of all Xi+x2-\- ■ ■ • + x„ with
x,G V. We now define, for some «>2, a function,/,
by
(6)
/ = F in V*,
/ = 0 outside V».
Now Vn is compact since -f- is a continuous function
pact space VX VX V ■ • ■ . Hence/has
compact support
on the comand so, since
FEL», we have
(7)
fELir\
Next consider the function
side Vn. We claim that
L1.
|/| 2—f(f*K).
it also vanishes
Clearly this vanishes out-
inside
Vn~x. For let xG Vn~x,
now unless K(x —t) —0 it follows that x —tE V and we conclude that
tEVn, so that f(t) = F(t). Thus fGf(t)K{x-t)dt = f0F(t)K(x-t)dt
for all xE F"-. Hence, for these x, \f\ 2-f(f*K) = \ F\*-F(F*K)=0
by hypothesis.
If we write A= Vn— Vn~x, then, we can sum up these remarks
(8)
Estimating
I/|2 -f(f*K) = 0 outside A.
|/| 2—f(f*K) inside A is our next task. Clearly
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in
632
D. J. NEWMAN
[August
f i/i»-/(ml
J A
I
(9)
F(x)¡I F(x- 0 II P(0 I dxdl.
eA,¡6F,x-ÍSF»
Note, however,
that the conditions
xG Fn_1 and ¿G F insure that
x —¿GF"-2. Combining this with x —¿GF" insures that x —¿GFn
- F"-2. Calling A'= F"- Vn~2, then, we may conclude that
Fix)Fix - t)K(t) | dxdt
i€i,¿ev,x-í6V"
(10)
Fix)Fix - t)K(f) | dxdt.
•J
>J T.CA>.
xeA',x-teA'
For each fixed t, however, Schwarz'
f f
(ID
inequality
| Fix)Fix-t)\dx£
»> J xeA'.l-iSA
*(=A' .z—t.(=A'
tells us that
f I Fix)|2dx
J A'
and so (10) yields
(i2) r r
i f(x)f(x- ojt(oi dxdt^ r i k i • r i f \
.tev.z-tev»
J JzeA.tev.x-teV"
Jq
Ja'
Combining (12) with (9) now yields
(13)
LiV-'til'K)
By virtue of (8), however,
(14) II Jfa \f\2-'fif*K)
Parseval's
(15)
<(1+/.lXl)/j'1'(13) can be written
ÚM \ \F\2,
where M=l+f|x|.
theorem is now justified by (7), and its application
I\Jgij/Kl-P)
ÚM JfA' \F\2.
Applying (5) to (15) allows the conclusion,
(i6)
and another
r i/i2 g if i «i JrA' if|2,
J G
application
of Parseval's
theorem yields
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gives
1965]
THEOREMSFOR CONVOLUTION-TYPE
EQUATIONS
(17)
f |/|2 = J fyn \F\^M\a\
633
f |F|2.
J Q
J &r
Finally (17) may be rewritten
(18)
I
J yn-2
Repeated
(19)
|F|2^ (1 -8) I I F |2, where 5 = —¡—fJyn
application
M \a \
of (18) now gives
f I F |2 g (1 - ô)"-1 f
J yi
I F |2 ^ (1 - S)n-1w(F2")
J y2n
(where m denotes Haar measure).
We now require the following:
Lemma. Let V be any compact subset of G. There exist constants c
and d such that m(Vn) ^cnd, for all n.
Proof. Let U be an open set containing V, such that its closure,
U, is compact. Since U2 is compact and since it is covered by the
totality
of translates
x+U,
it follows that
i/2CUf=0 (xi+U),
for
some finite collection of x¿. Hence c72CUf=0 (x,+ i7) and, by induction, UnCUj (yj-r U), where the y¡ run through all the possible
choices of Xjj+Xij-r- • • • +x,n_,. Since the number of such choices is
/n + d - 1\
V d
)'
however, we conclude that
in + d - 1\
m(Vn) g m(Un) g (
Jm(U).
This last expression is ^cnd for an appropriate choice of c, however, and the proof is complete.
If we now combine the result of this lemma with (19) above we
conclude that
(20)
f |F|2^(l-5)**-1cO)<i,
J yi
and letting n—>«>tells us that fy*\ F| 2= 0, or
(21)
F = 0 a.e.,
in V2.
But, for xEV, F*K=fvtF(t)K(x-t)dt
and so, by (21), F*K = 0
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634
AZARIA PAZ
[August
throughout V. By hypothesis, however, | F\2 = F(F*K), and so F = 0
identically throughout V.
In particular, then, F vanishes identically in a neighborhood of x0.
Since x0 was arbitrary F vanishes identically and the proof is com-
plete.
Yeshiva University
DEFINITE AND QUASIDEFINITE SETS OF
STOCHASTIC MATRICES'
AZARIA PAZ
1. Introduction. This note is concerned with asymptotic behaviour
of long products of stochastic matrices of a given form. Its objects are:
(a) To prove that a theorem stated (but not proved) by the author
in a previously-published
paper [2] is equivalent to one proved by
Wolfowitz in [l].
(b) To formulate a decision procedure for the above
preferable to that given by Wolfowitz in [l].
(c) To solve a related problem.
Familiarity with the above two papers is desirable.
problem,
2. Definitions.
(We adopt here some of the definitions used by
Wolfowitz.) A finite square matrix P = ||^,,|| is called stochastic if
pa^O for all i, j and 53 jpu=l
A stochastic
matrix
for all i.
P is called indecomposable
and aperiodic
(S.I.A.) if
Q = lim P"
n—»00
exists and all rows of Q are the same. |P|
and 8(P) are defined as
I PI = max I PaI ,
d(P) = max max | phi — piti \ .
i hh
With every stochastic matrix P we associate a finite graph having
ra states (vertices)—ra being the order of P—such that transition is
Received by the editors April 26, 1964.
1 The note is based on part of the author's D.Sc thesis submitted
the Technion, Israel Institute of Technology, December 1963.
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to the Senate of